the large sphere has a diameter of 20 feet. which expression represents the volume, in cubic units, of the…

the large sphere has a diameter of 20 feet. which expression represents the volume, in cubic units, of the shaded part of the sphere? \\(\\frac{4}{3}\\pi(10^3) + \\frac{4}{3}\\pi(4^3)\\) \\(\\frac{4}{3}\\pi(10^3) - \\frac{4}{3}\\pi(4^3)\\) \\(\\frac{4}{3}\\pi(20^3) + \\frac{4}{3}\\pi(4^3)\\) \\(\\frac{4}{3}\\pi(20^3) - \\frac{4}{3}\\pi(4^3)\\)

the large sphere has a diameter of 20 feet. which expression represents the volume, in cubic units, of the shaded part of the sphere? \\(\\frac{4}{3}\\pi(10^3) + \\frac{4}{3}\\pi(4^3)\\) \\(\\frac{4}{3}\\pi(10^3) - \\frac{4}{3}\\pi(4^3)\\) \\(\\frac{4}{3}\\pi(20^3) + \\frac{4}{3}\\pi(4^3)\\) \\(\\frac{4}{3}\\pi(20^3) - \\frac{4}{3}\\pi(4^3)\\)

Answer

Explanation:

Step1: Find large sphere radius

The diameter of the large sphere is 20 ft, so radius $r_1 = \frac{20}{2} = 10$ ft. Volume of large sphere: $V_1 = \frac{4}{3}\pi r_1^3 = \frac{4}{3}\pi(10^3)$

Step2: Find small sphere radius

The small sphere has radius $r_2 = 4$ ft. Volume of small sphere: $V_2 = \frac{4}{3}\pi r_2^3 = \frac{4}{3}\pi(4^3)$

Step3: Calculate shaded volume

Shaded volume = Large sphere volume - Small sphere volume $V_{\text{shaded}} = V_1 - V_2$

Answer:

$\boldsymbol{\frac{4}{3}\pi(10^3) - \frac{4}{3}\pi(4^3)}$ (the second option)